Sounds of Manifolds

Carolyn Gordon has a paper in the Mathematical
Intelligencer (1989, vol 11, no. 3, pp. 39 - 47) entitled "When
You Can't Hear the Shape of a Manifold." Accompanying this article
are a sidebar and floppy vinyl record by Dennis DeTurck. Reproduced
here, with Dr. DeTurck's permission, are the tracks from the record
in .mp3 format; I've included his comments from the sidebar with each
track. Two tech-tips: (1) you might need to turn the volume up (I
forgot to amplify them when I was editing them) and (2) if you play
more than one of these at once, it sounds really weird.

However, I'm not going to discuss the mathematics on this
page.

Track 1:
Your browser does not support the audio element.

The "harmonic sequence" of the circle S1 (e.g., sine
tones whose frequencies are integer multiples of the first one
heard). Then we build a complex S1 -tone (e.g., beginning
with the fundamental tone, more and more additional frequencies are
mixed in, based on the circle's eigenvalues). Then the complex tone
is repeated, with other characteristics added (attack and decay
envelope) to get an "electronic piano" sound. As an example, the
Dresden Amen is played on this "circular" piano.

Track 2:
Your browser does not support the audio element.

The analog of 1 for the two-sphere S2: the "harmonic
sequence" of S2 (just like 1, but with the sphere's
eigenvalues), then we build a complex S2 -tone, repeat it
with the piano's attack-decay envelope, and perform the on the
"spherical piano."

Track 3:
Your browser does not support the audio element.

The effect of dimension: A simple piano exercise is repeated,
first on S1, then on S2, then on S3
(which has the same eigenvalues, but not multiplicities, as the
complex projective plane C P2), then on
S6 (same as the quaternionic projective plane H
P2 ), and then on S12 (same as the Cayley plane
Ca P2 ).

Track 4:
Your browser does not support the audio element.

The "Romanza" movement, from Beethoven's Sonatina in G is
performed on various "spherical pianos." The first section is played
on an S1-piano, the second on S2, the next on
S3, the next on S6, and a codetta back on
S1.

Track 5:
Your browser does not support the audio element.

A chorus of projective planes: The "Bouree" from Handel's
Fourth Flute Sonata is performed with C P2
playing the flute part, Ca P2 doing the bass line,
and H P2 providing the realization (middle
parts).

Track 6:
Your browser does not support the audio element.

The piano exercise is played on S1 again, then on
various flat rectangular tori. The first is the square torus
T1:1, then on tori whose side ratios are 3:5, 11:13,
21:23, and finally 31:33. (Hear the effect of the ratio being close
to but not equal to 1?)

Track 7:
Your browser does not support the audio element.

Torus music: Robert Schumann's Wild Rider is performed on a
piano whose strings are T3:5.

Track 8:
Your browser does not support the audio element.

More torus music: J. S. Bach's Two-part Invention No. 4 in d
minor is performed on a T11:13 piano.

Track 9:
Your browser does not support the audio element.

Spheres and tori together: The "Presto" movement from Scarlatti's
Sonata in C (Longo S.3) is performed, with the upper voice
played on T21:23 and the other parts on Ca
P2.

Track 10:
Your browser does not support the audio element.

A "one-minute quiz": Chopin's Valse in D-flat,Opus 64, No.
1 ("Minute Waltz"). The oom-pahs are performed on a standard
S1-piano. Can you identify the seven manifolds that play
different sections of the tune?

In the Intelligencer, answers were provided on p. 79.
However, I didn't look at that page, so if you need to know the
answers... for now, you'll have to go to the library.

Tech thanks to Jon Fretheim for the embedded-player code and to
Sean Kinlin & UNI Electronic Media for letting me use their
sound-editing equipment.